How do i snap to an intersection of the edge when locking an axis?
I don't know if there is a better alternative but you could try multi point snap.

Simply press A while you are snapping to create a snap point. Add multiple points to or near the location you want to snap to to increase the influence of where your vertex will snap. In this case adding a point below and above where the Y axis crosses the edge will give you a close approximation, requires some fiddling to get used to but should be useful.